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definite order, corresponding to 'numberable' properties of the
elements, the assignment in the given order corresponding to a
preassigned ordering of the 'numberable* properties. Granted
that this may be even less comprehensible than Riemann's
definition, it is nevertheless a -working basis from which to
start, and all that it amounts to in plain mathematics is this:
a manifold is a set of ordered 'n-tuples' of numbers (aj1Ja?s ... 5
#), where the parentheses, (), indicate that the numbers
lsa22, ..., xn are to be written in the order given. Two such
tt-tuples, (o!lyr2,... , xn) and (y-&& ... , yn) are equal when, and
only when, corresponding numbers in them are respectively
equal, namely, when, and only when, a^ = ylt #2 = yzi ..,}
*n = !/
If precisely n numbers occur in each of these ordered n-tuples
in the manifold, the manifold is said to be of n dimensions. Thus
we are back again talking co-ordinates with Descartes. If each
of the numbers in (ajlsa2, ... , xn) is a positive, zero, or negative
integer, or if it is an element of any countable set (a set whose
elements may be counted off 1,2,3, ...), and if the like holds for
every n-tuple in the set, the manifold is said to be discrete. If
the numbers xvx, ... , xn, may take on values continuously
(as in the motion of a point along a line), the manifold is
This working definition has ignored - deliberately - the ques-
tion of whether the set of ordered n-tuples is 'the manifold' or
whether something "represented by' these is 'the manifold'.
Thus, when we say (x,y) are the co-ordinates of a point in a
plane, we do not ask what 'a point in a plane' is, but proceed to
work with these ordered couples of numbers (x%) where x,y run
through all real numbers independently. On the other hand it
may sometimes be advantageous to fix our attention on what
such a symbol as (x,y) represents. Thus if # is the age in seconds
of a man and y his height in centimetres, we may be interested
in the man (or the class of all men) rather than in his co-ordinates
with which alone the mathematics of our enquiry is concerned. In
this same order of ideas, geometry is no longer concerned with
what 'space' 'is' -whether 'is' means anything or not in relation
to 'space1. Space, for a modern mathematician, is merely a